As an experiment, I tried to prove this using
$$
(0) \;\;\; |p| = p \max -p
$$
together with the properties of $\;\max\;$ (and $\;\min\;$). The result is not too pretty, but the upside is that I could do it almost mechanically, and without any case distinctions.

for all $x,y \in \mathbb{R}$ $|x-y|+|y| \ge|x-y+y|$(by triangular inequality) and done the proof. In fact, by symmetry, nomatter the case $x>y$ or $y>x$ would also yield the inequality so $|x-y| \ge ||x|-|y||$.